In this short note we give a polynomial-time quantum reduction from the vectorization problem (DLP) to the parallelization problem (CDHP) for efficiently computable group actions.
Combined with the trivial reduction from parallelization to vectorization, we thus prove the quantum equivalence of these problems, which is the post-quantum counterpart to classic results of den Boer and Maurer in the classical Diffie-Hellman setting. In contrast to the classical setting, our reduction holds unconditionally and does not assume knowledge of suitable auxiliary algebraic groups.
We discuss the implications of this reduction for isogeny-based cryptosystems including CSIDH.

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